Contributions à l'Etude du Vertex Topologique en Théorie ... - Toubkal

L.B. Drissi et al. / Nuclear Physics B 804 [PM] (2008) 307–341 311Fig. 1. The toric web-diagram of C 3 . It appears as local patches in toric Calabi–Yau threefolds. The three edges end onstacks of Lagrangian D branes. λ, μ and ν are 2d partitions which, in QFT set up, may be thought of as the externalmomenta.(iii) The boundary conditions (open strings) described by 2d partitions μ (generic representationsof U(∞)). In the QFT language where Feynman graphs play a quite similar roleas the toric web-diagrams, the 2d partition μ corresponds to the “external momentum” ofFeynman graph. Recall that a 2d partition μ is a Young diagram with columns(μ 1 ,μ 2 ,...), μ i μ i+1 , μ i ∈ Z + .Columns of the 2d partition are associated with Lagrangian D-branes and rows with Lagrangiananti-D-branes.(iv) Lagrangian D-brane/anti-D-brane pairs are needed for the gluing of the vertices. The gluingoperation is achieved by inserting 2d partitions ν and their transpose ν T at the cuts andsumming over all possible ν’s. In QFT language, ν corresponds to “internal momenta”.The topological 3-vertex 3 method for computing the partition function Z X3 (q) is illustrated onthe three examples given below.2.2. ExamplesExample 1 (The 3-vertex of C 3 ). The toric graph of C 3 is given by Fig. 1. Following [28], thepartition function of the 3-vertex, with a stack of Lagrangian D-branes ending on its legs capturedby the boundary conditions (λ,μ,ν), is given byZ X3 (q) = ∑C λμν (q)(Tr λ V Tr μ V Tr ν V).(2.2)λ,μ,νIn this relation, the trace Tr λ of the holonomy matrix V, with eigenvalues x = (x 1 ,x 2 ,...),isgiven by the Schur function S λ (x). The latter depends on the 2d partition λ = (λ 1 ,λ 2 ,...) andthe x i = q i−1/2−λ i. The rank three tensorC (3) = C λμν ,is the topological 3-vertex whose explicit expression reads asC λμν (q) = q[S κ(λ) (ν T q−ρ ) ∑with ρ = (ρ 1 ,ρ 2 ,...)and ρ i = 1/2 − i.2d partitions η(S λ T /η q−ν−ρ ) (S μ/η q−ν T −ρ )](2.1)(2.3)(2.4)3 For simplicity, we use 3-vertex to refer to the planar 3-vertex.